Лебединская. Динамика материальной точки
.pdf
' 1.11
0 0 7
8 : + 7 ; :
( III ± )
9 &) # ! ,: 1) 4
4- /, ,
-4 ; 2) ( . +# !#, # & . !: /, ,
, (, .
( ! * !'* # %
/, , . 5.1. /
* * R1 R2,
* / ( . 11.1
|
|
). |
- |
|
|
|
m0 , - |
||
|
|
**l 4 |
||
|
|
. |
, |
|
|
|
. |
||
|
|
1 ( |
||
|
|
m, |
|
|
|
|
/ |
4- |
|
|
|
. ' h, |
||
|
|
t, |
||
|
|
. |
|
|
|
|
/ 4 - |
||
|
|
|
||
|
|
|
M = I ε , |
(11.1) |
, |
'. 11.1 |
' – 4-4 , -4 |
||
, I – ( , ε – .(11.1) ,
( (,
I1ε1 = I2ε2 , |
|
ε2 / ε1 = I1 / I2 . |
(11.2) |
|||
( -4 |
||||||
. ( |
|
( |
||||
I0 ( * I ′: |
||||||
I |
= I0 + 4I ′. |
|
|
(11.3) |
||
( 3 |
||||||
′ |
′ |
|
2 |
|
|
|
I |
= I0 + m0l |
, |
(11.4) |
|||
|
||||||
63
I |
′ |
– ( , * 4 |
0 |
( 4 . , (11.4) (11.3), :
′ |
2 |
|
|
|
I = I0 + 4I0 + 4m0l |
. |
(11.5) |
||
|
(11.5) ** -
4: * l 1
4, (
|
′ |
2 |
|
|
I1 = I0 + 4I0 + 4m0l |
1 , |
(11.6) |
* l 2 , ( |
|
||
|
′ |
2 |
|
|
I2 = I0 + 4I0 + 4m0l |
2 . |
(11.7) |
′ |
, , - |
||
I0 I0 |
|||
I1/I2, , (11.2).
(11.1). . :
( , *
2, .
+ ( *,
(11.6) (11.7)
I1 − I2 = 4m0 (l 12 − l 22 ) . (11.8)
6 ( ,
(11.1)
I1 − I2 |
= M1 − M 2 . |
(11.9) |
|
|
ε1 |
ε2 |
|
! 2 - , 24-4
F . 1- *,
m. II -
r |
r |
|
mg + F |
= ma , |
|
mg − F = ma . |
(11.10) |
|
(11.10), - 4-4
M = m(g − a)R . |
|
(11.11) |
& |
||
h = at 2 , a = |
2h . |
(11.12) |
2 |
t2 |
|
, , * -
* R - . !,
ε = a / R . |
(11.13) |
, (11.11), (11.13) (11.12), (11.9) ( 2
(11.11) - a ,
64
g), - 2
( **
I |
1 |
− I |
2 |
= mR2g (t2 |
− t2 ) . |
(11.14) |
|
|
|
2h |
1 |
2 |
|
||
|
|
|
|
|
|
|
|
/, (11.1)
(11.8) (11.14),
4m |
(l 2 |
− l 2 ) = mR 2g |
(t 2 |
− t2 ) , |
(11.15) |
|
0 |
1 |
2 |
2h |
1 |
2 |
|
|
|
|
|
|
|
|
t1 t2 – * m h |
* |
|||||
4l 1 l 2 .
, 4 m0 4
(11.15), ,
4 .
+!# ! , !& # ! ,
1.8 m0 * 4 l 1 ( -
) .
2., m
t h.
3., m0 l 2 3 .
4.' (
+ |
m0 |
m |
R |
h |
l 1 |
t1 |
< t1> |
l 2 |
t2 |
< t2> |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5. , (11.15).
! #!&) ,! #! ,
1.% ( , .
2.% . 1 ?
3.% , . ,
2 * . 1 * ?
* ( * - ?
4.! 3 . 8 .
5.4 .
6.1 - M ε ? (11.15).
m0 = 200 83 ( ).
# (#
1.! . . 1 . +. 1. – .: , 1989. – !. 94–116.
2.+ +. . 1 . – .: . ., 2001. – !. 34–46.
65
' 1.12
+ + ++ 7 ; :
9 &)# ! ,: .
+# !#, # & . !: , -
, .
( ! * !'* # %
& ( . 12.1)
,
, , 2 , 2 , * .
(12.2). / 2, ( * - 4
66
. , - 4 3. &
1, , - 4
&&. , 4-4 3
( .
'. 12.2
, « »,
« » .
2 « » fmax , *. , (
« ». , ,
« » « » ,
, *
( * . ,
T , 4-
t , n
T = nt .
! ² – ² , -
, * . +
- (
), * ² – ²
mυl = (I +ml 2 )ω0 , |
(12.1) |
m – , υ – , l |
– - |
4 &&, ω0 –
, I – ( 4. , ( ml 2 -
( , (12.1) -
,
υ = |
I ω0 |
. |
(12.2) |
|
|||
|
ml |
|
|
67
(12.2) : υ ,
, ω0 , (
I . + , *4 .
* * 2. 0 *,
2
(2 :
I ω02 |
= |
Dφmax2 |
, |
(12.3) |
2 |
|
|||
2 |
|
|
||
D – 2 ( , φmax – -
.
- * ( . 1.3):
T = 2π |
|
I |
|
. |
(12.4) |
|
|||||
|
|
D |
|
||
(12.4) - 2 ( D,
- (12.3) -
ω0 . , ω0 (12.2) - -
υ = 4π2I φmax . |
(12.5) |
ml T 2 |
|
(12.5) - ,
( I. , 3
( . 8 (
, ( * * R1 R2 4:
I |
= I |
0 |
+ 2MR2 |
, |
|
1 |
|
1 |
|
|
|
I2 = I0 + 2MR22 , |
(12.6) |
||||
I0 – ( * -
( * , M – .
(12.6), * ,
I1, I2, *
I = I1 − I2 ,
I = 2M (R2 |
−R2 ) . |
(12.7) |
1 |
2 |
|
% * * ( - (12.4). 8- (12.4) * 3
T1 = 2π
DI1 ,
68
T2 = 2p |
|
I2 |
|
. |
(12.8) |
|
|||||
|
|
D |
|
||
8, (12.8) ( ,
I2 = I1 + |
I , - |
|
|
|
|
|
|
|
|
|
|
|
T 2 |
DI . |
(12.9) |
||
|
|
|
I1 = T 2 -T 2 |
|||||
|
|
|
|
|
1 |
|
|
|
|
|
|
2 |
1 |
|
|
|
|
, (12.9) (12.7) (12.5), - |
|
|||||||
|
|
|
|
|
||||
|
υ = |
4πφmaxT1 |
M (R2 |
-R2 ) , |
(12.10) |
|||
|
|
|||||||
|
|
(T 2 |
-T 2 )ml |
|
2 |
1 |
|
|
|
|
2 |
1 |
|
|
|
|
|
T1 , |
T2 , fmax l – , |
* |
||||||
, fmax ( I1. |
|
|||||||
+!# ! , !& # ! ,
1.3 4 *
. 4 ( R1 . %
R1
.
2.8 . % *
* -, 2
.
3.&, * ,
² ², * . , *
φmax .
4. / *1. % 2
t1 , 10–15 * .
5./ l ′
.
6.% R1 2, 3, 4, 5 * .
7., 4 R2 .
8.% R2 * 2, 4 5
* .
9./ 4
l : l = L − l ′ , L – 4
. % L
. 10.' (
69
n R1 t1 T1 <T1> R2 t2 T2 <T2> l ′ <l ′ > φmax <φmax>
1
2
3
11./ , . &,
= 168 .
12., * 4- (12.9)
.
13./(
* , ,
²' … ²
! #!&) , ! #! ,
1.9 ( ? 9 * ?
*( * ?
2.9
? 1 * ?
3.8 2 4-4 - ( 2 .
4.1 * * 2
?
5.* (
1.3).
6.1
(12.9)?
7.8 * ² –
².
8.8 *2 ² – ².
9.(12.9) .
# (#
1.! . . 1 . +. 1. – .: , 1989. – !. 94–116.
2.+ +. . 1 . – .: . ., 2001. – !. 34–46.
3.8 . .., + /. . 1 . +.1. – .: , 1972. – !. 59–70.
70
' 1.13
+>== 9: 8 :
: : + 7 ; :
9 &) # ! ,: ;
2 ( .
+# !#, # & . !:
; – (, , .
#!#
*** *( * - . *
- 4 * 2 -- 2- ( -).
'- $ , -4 *
* * 4, ( # , -4 4
( - ²
²). .
4 -
, -
- * ( )
( ). 1
*- -42 (. ? -
. , , 4
, , . 6 ,
, -4 - –
. + , 4
-4, 4 . !
. ((
. 8,
. 1 ,
- .
+ |
|
||
1 2 ( μ - F |
|
||
( N ( , III - , |
|||
F , -4 ) |
|
||
m = |
F |
. |
(13.1) |
|
|||
|
N |
|
|
% 2 (
( . 13.1). 3 ,
&, - - ,
b . 0
71
α0, *-4 .
n αn . 8 * *
.
.
@
/
|
|
'. 13.1
( . 13.1 ) ( -
F , -4 mg ,
N = mgcosβ, |
(13.2) |
m – , g – .
! 4- *2.
, *2, ,
n , - (2
F S = mg h , |
(13.3) |
F S = A – , S – , |
n |
, mg h = E – ( 2
, h – * . ! (13.2) (13.3) (13.1) :
μ |
= |
|
h |
. |
(13.4) |
|
S cosβ |
||||||
|
|
|
|
|||
%, . 13.1 , |
|
|
|
|
|
|
|
h = |
l sinβ, |
(13.5) |
|||
l – ,
.
72